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Chapter 2: Problem 59

Solve using the multiplication principle. Don't forget to check! $$ \frac{1}{9}=\frac{z}{5} $$

Short Answer

Expert verified
z = \(\frac{5}{9}\).

Step by step solution

01

Understand the Equation

The given equation is \(\frac{1}{9}=\frac{z}{5}\). This is an equation involving two fractions. To solve for \z, we can use the multiplication principle.
02

Apply the Multiplication Principle

To isolate \z, multiply both sides of the equation by 5, which is the denominator of the fraction containing \z. This gives us: \(\frac{1}{9} \times 5 = \frac{z}{5} \times 5 \).
03

Simplify the Equation

Simplifying both sides, we get: \( \frac{5}{9} = z \). Therefore, \z = \frac{5}{9}.
04

Check the Solution

To ensure that \(z = \frac{5}{9}\) is correct, substitute \z back into the original equation: \(\frac{1}{9} = \frac{\frac{5}{9}}{5}\). This simplifies to: \(\frac{1}{9} = \frac{5}{45}\), and since \( \frac{5}{45} \) reduces to \( \frac{1}{9} \), the solution checks out.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving equations
Solving equations is one of the most fundamental skills in algebra. An equation is a mathematical statement that asserts the equality of two expressions. To solve an equation means to find the value of the variable that makes the equation true.
The process typically involves isolating the variable on one side of the equation.
In our example, we started with: \[^ \frac{1}{9}=\frac{z}{5} \]
To isolate \(z\), we applied the multiplication principle.
fractions
Fractions represent parts of a whole.
A fraction consists of a numerator (top number) and a denominator (bottom number).
In the given exercise, we worked with the fractions \( \frac{1}{9} \) and \( \frac{z}{5} \).
When solving equations with fractions, it's often useful to eliminate the fractions by finding a common denominator or by multiplying both sides of the equation by a value that will clear the fractions, as done in our solution.
algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
In the exercise provided, we used basic algebraic principles to solve for \(z\).
We applied the multiplication principle to both sides of the equation, which is essential in solving linear equations involving fractions.
This method keeps the balance of the equation while isolating the variable, allowing us to determine its value systematically.

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Most popular questions from this chapter

Use an inequality and the five-step process to solve each problem. Abriana rented a compact car for a business trip. At the time of the rental, she was given the option of prepaying for an entire tank of gasoline at \(\$ 3.099\) per gallon, or waiting until her return and paying \(\$ 6.34\) per gallon for enough gasoline to fill the tank. If the tank holds 14 gal, how many gallons can she use and still save money by choosing the second option?

Use an inequality and the five-step process to solve each problem. All Seasons Well Drilling offers two plans. Under the "pay-as-you-go" plan, they charge \(\$ 500\) plus \(\$ 8\) per foot for a well of any depth. Under their "guaranteed-water" plan, they charge a flat fee of \(\$ 4000\) for a well that is guaranteed to provide adequate water for a household. For what depths would it save a customer money to use the pay-as-you-go plan?

Use an inequality and the five-step process to solve each problem. In order to qualify for a batting title, a major-league baseball player must average at least 3.1 plate appearances per game. For the first nine games of the season, a player had \(5,1,4,2,3,4,4,3,\) and 2 plate appearances. How many plate appearances must the player have in the tenth game in order to average at least 3.1 per game?

Graph the solutions of \(|x|<3\) on the number line.

A storekeeper goes to the bank to get \(\$ 10\) worth of change. She requests twice as many quarters as half dollars, twice as many dimes as quarters, three times as many nickels as dimes, and no pennies or dollars. How many of each coin did the storekeeper get?
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